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Conditional partial exchangeability: a probabilistic framework for multi-view clustering

Beatrice Franzolini, Maria De Iorio, Johan Eriksson

TL;DR

This work addresses multi-view and longitudinal clustering by introducing conditional partial exchangeability (CPE), a principled way to couple dependent partitions across features while preserving subject identities. It develops the telescopic clustering framework, with two concrete instantiations: an infinite-label t-HDP model and a random-label telescope using a unique-atom process, both satisfying CPE and offering tractable inference. The authors define dependence measures (tau, t-EPPF, TARI) to quantify cross-layer relationships and provide posterior-inference algorithms, along with simulation studies and a real-data application to childhood obesity that reveal interpretable, cross-layer associations among growth trajectories, maternal factors, and metabolite profiles. The framework enables flexible, computationally feasible dependent partition models for multi-view data and suggests future work on broader polytree structures and theoretical questions about the necessity of CPE for identity preservation.

Abstract

Standard clustering techniques assume a common configuration for all features in a dataset. However, when dealing with multi-view or longitudinal data, the clusters' number, frequencies, and shapes may need to vary across features to accurately capture dependence structures and heterogeneity. In this setting, classical model-based clustering fails to account for within-subject dependence across domains. We introduce conditional partial exchangeability, a novel probabilistic paradigm for dependent random partitions of the same objects across distinct domains. Additionally, we study a wide class of Bayesian clustering models based on conditional partial exchangeability, which allows for flexible dependent clustering of individuals across features, capturing the specific contribution of each feature and the within-subject dependence, while ensuring computational feasibility.

Conditional partial exchangeability: a probabilistic framework for multi-view clustering

TL;DR

This work addresses multi-view and longitudinal clustering by introducing conditional partial exchangeability (CPE), a principled way to couple dependent partitions across features while preserving subject identities. It develops the telescopic clustering framework, with two concrete instantiations: an infinite-label t-HDP model and a random-label telescope using a unique-atom process, both satisfying CPE and offering tractable inference. The authors define dependence measures (tau, t-EPPF, TARI) to quantify cross-layer relationships and provide posterior-inference algorithms, along with simulation studies and a real-data application to childhood obesity that reveal interpretable, cross-layer associations among growth trajectories, maternal factors, and metabolite profiles. The framework enables flexible, computationally feasible dependent partition models for multi-view data and suggests future work on broader polytree structures and theoretical questions about the necessity of CPE for identity preservation.

Abstract

Standard clustering techniques assume a common configuration for all features in a dataset. However, when dealing with multi-view or longitudinal data, the clusters' number, frequencies, and shapes may need to vary across features to accurately capture dependence structures and heterogeneity. In this setting, classical model-based clustering fails to account for within-subject dependence across domains. We introduce conditional partial exchangeability, a novel probabilistic paradigm for dependent random partitions of the same objects across distinct domains. Additionally, we study a wide class of Bayesian clustering models based on conditional partial exchangeability, which allows for flexible dependent clustering of individuals across features, capturing the specific contribution of each feature and the within-subject dependence, while ensuring computational feasibility.
Paper Structure (21 sections, 12 theorems, 55 equations, 6 figures, 1 table)

This paper contains 21 sections, 12 theorems, 55 equations, 6 figures, 1 table.

Table of Contents

  1. Clustering multi-view information
  2. Conditional partial exchangeability
  3. The class of telescopic clustering models
  4. A general telescopic clustering model
  5. Measures of telescopic dependence
  6. Extension to $L$ layers using polytrees
  7. A telescopic model with infinite number of labels
  8. A telescopic model with random number of labels
  9. Algorithms for posterior inference
  10. Numerical studies
  11. Simulation study
  12. An application to childhood obesity
  13. Conclusions and future directions
  14. Proof of Theorem 1
  15. Proof of Proposition 1
  16. ...and 6 more sections

Key Result

Theorem 1

If $(X_{2i})_{i\geq1}$ is conditionally partially exchangeable with respect to $\rho_{1}$, then, for any measurable $A$ where a strict inequality in s-i) is achievable as long as $(X_{2i})_{i\geq1}$ is not conditionally exchangeable with respect to $\rho_1$.

Figures (6)

  • Figure 1: Toy example. Data was simulated for two features and 200 subjects, with the true clustering configuration displayed in panel (a). The first feature (x-axis) is sampled from a univariate Normal with unitary variance and mean equal either to 1 or -1, depending on the true cluster assignment. The second feature (y-axis) is sampled from a three-variate Normal with identity covariance matrix and mean equal to either $(1,1,1)$ or $(-1,-1,-1)$ depending on the true cluster assignment; it is represented via its first principal component on the y-axis. The Dirichlet process estimate in panel c) is obtained with a Multivariate Normal kernel with identity covariance matrix, standard Multivariate Normal base measure, and concentration parameter equal to 0.1. Both the clustering configurations obtained with k-means (panel b) and the Dirichlet process mixtures (panel c) are heavily informed by the second feature and ignore the information contained in the first feature.
  • Figure 2: Layer dependence for longitudinal data.
  • Figure 3: Triangular dependence for three layers.
  • Figure 4: Scenario 2. Rand indexes between the truth and the estimated configuration.
  • Figure 5: Simulation study: results for Scenario 2. Pairwise Rand indexes between any couple of layers for (a) the true clustering configurations; (b) the t-HDP model; (c) k-means.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Definition 1: Conditional partial exchangeability
  • Theorem 1: Subjects' identity across layers
  • Proposition 1: Temporal random partition model
  • Proposition 2: Separately exchangeable NDP-CAM
  • Proposition 3: Dependent Dirichlet processes
  • Definition 2: Telescopic clustering model
  • Theorem 2: Telescopic clustering - joint representation
  • Definition 3: Measure of telescopic dependence
  • Proposition 4: Dependence measures as functions of the number of clusters
  • Definition 4: Telescopic adjusted Rand index
  • ...and 15 more